Tight Convergence Rates in Gradient Mapping for the Difference-of-Convex Algorithm

TL;DR

This paper provides new theoretical convergence guarantees for the difference-of-convex algorithm (DCA), including tight sublinear rates in various curvature regimes, extending existing results and connecting to proximal gradient descent.

Contribution

It introduces a comprehensive analysis of DCA with weakly-convex functions, establishing tight convergence rates across multiple parameter regimes and linking DCA to PGD.

Findings

Six parameter regimes with sublinear convergence rates identified.

Recovery of existing rates for standard decompositions.

New rates established for non-standard curvature settings.

Abstract

We establish new theoretical convergence guarantees for the difference-of-convex algorithm (DCA), where the second function is allowed to be weakly-convex, measuring progress via composite gradient mapping. Based on a tight analysis of two iterations of DCA, we identify six parameter regimes leading to sublinear convergence rates toward critical points and establish those rates by proving adapted descent lemmas. We recover existing rates for the standard difference-of-convex decompositions of nonconvex-nonconcave functions, while for all other curvature settings our results are new, complementing recently obtained rates on the gradient residual. Three of our sublinear rates are tight for any number of DCA iterations, while for the other three regimes we conjecture exact rates, using insights from the tight analysis of gradient descent and numerical validation using the performance estimation methodology. Finally, we show how the equivalence between proximal gradient descent (PGD) and DCA allows the derivation of exact PGD rates for any constant stepsize.